In this section, we calculate the sensory measurement error of reward magnitude e. In some modalities, the sensory receptor is itself thought to produce scalar noise Matthews et al.
While this is possible in the measurement of reward magnitude, we do not consider this simple solution here as neural elements in the central nervous system are typically considered to approximate Poisson processes, which have square-root noise and not linear noise Rieke et al.
Rather, we consider errors in ascribing value to a given reward magnitude as resulting from central and not peripheral processes.
While there are other models for Weber's law in sensation Treisman, ; Dehaene, ; Deco and Rolls, ; Shouval et al. To this end, we assume that the sensory process for measuring the magnitude is carried out in time over a small temporal window of sensation. This sensory window is defined as the time over which there is a constant rate of sensory input.
Hence, we assume that the net perceived reward magnitude is proportional to the time it takes to integrate the sensory input e. For an alternative model of sensory integration, see Appendix A1 in Supplementary Material.
In order to evaluate the noise in measurement, we assume that this sensory integration can be described by an accumulator model similar to previous decision-making models used for evidence accumulation e. We further assume that the reward magnitude is represented linearly and does not undergo a logarithmic transformation, as has been suggested for number representation Dehaene, In the rest of this section, we formalize this accumulator model using a stochastic differential equation, and then analytically calculate the time dependence of its mean and variance.
If the neural system carrying out this sensory integration were perfectly noiseless, we can describe the accumulator model by the following differential equation. Here, r t represents the integrated reward magnitude at a given time. Thus, the measured reward magnitude r will be the integrated magnitude at the end of the sensory window, t sensory , i.
The rate of sensory input is denoted by a. We now relax the assumption that the sensory integration is noiseless.
Noise in such an accumulator system can result from two sources: noise in the sensory input and feedback noise in the accumulator. Thus, the variance of the sensory input will be proportional to the input a and the feedback noise will be proportional to r t. For simplicity, we first assume that these two sources of noise are independent and additive. Thus, the introduction of these noise sources can be formally described by the following stochastic differential equation.
We will analytically solve the time dependence for the first and second moments of the above accumulator [shown in Equation 11 ] so as to calculate the mean and variance at the end of the sensory window. Since the measured reward magnitude is the integrated magnitude after the sensory window, the CV of the measurement can be written as. If one assumes that the rate of sensory input is a constant, the above equation shows that except for low reward magnitudes, the CV is a constant, i.
These analytical results are confirmed in numerical simulations as shown in Figure 4. Figure 4. Confirmatory simulations see Methods of the analytical solution of an accumulator model in which the sensory and feedback noise combine additively.
The black dots show the results of numerical simulation. The results approximate Weber's law well but for low reward magnitudes and high sensory noise b. The mathematics of the accumulator shown in Equation 11 is quite similar to Equation 9 in Simen et al.
But there are some significant differences in the meaning of the terms. First, our model is for reward magnitude perception, whereas theirs is for time interval production.
Second, as a consequence, while in our model the rate of sensory input is assumed to be a constant, they assume that the rate of accumulation is tuned for the interval to be timed. Equations 11—21 assumed that the sensory input noise is additive with respect to the feedback noise.
Instead, if this noise were in fact multiplicative, Equation 11 would change to. Thus, when the sensory and feedback noises multiply, the coefficient of variation is independent of the magnitude of the sensory signal a. Again, we performed confirmatory numerical simulations of Equation 22 , the results of which are shown in Figure 5. Therefore, if the sensory input noise is multiplicative, the coefficient of variation is exactly constant, thus making Weber's law exact.
Instead, if the sensory input noise is additive, the coefficient of variation shows deviations from exact Weber's law at low reward magnitudes. Figure 5. Confirmatory simulations see Methods of the analytical solution of an accumulator model in which the sensory and feedback noise combine multiplicatively. Here, Weber's law is exact. The accumulator model considered above is similar to the one that we previously proposed for the representation of subjective time Namboodiri et al.
The most important difference is that whereas subjective time is assumed to be a non-linear transform of real time, subjective reward is assumed to be linearly proportional to the real reward. Due to this difference, the reward magnitude accumulator is analytically tractable, unlike the subjective time accumulator, for which the analytical solution was approximate Namboodiri et al.
The other difference is that since the reward magnitude accumulator operates on a sensory input unlike the subjective time accumulator , the contribution of this sensory noise has also been included. We now have all the elements to calculate the error in subjective value of a delayed reward resulting from errors in both magnitude and time measurements Figure 6.
Figure 6. The error in subjective value is affected by errors in the measurement of both delay as shown in Figure 2 and reward magnitude. This combined error is calculated analytically in Section Combined error due to time and magnitude measurements on subjective value. Since we are only interested in the net error, so as to match the direction of change, we will consider the effect of error in both r and ST t by adding the JND of r and subtracting the JND of ST t.
Thus, we get the following equation. Therefore, Equation 25 becomes. For simplicity, we consider the exact form of Weber's law to hold for the sensory measurement of r. From Equations 6 , 8 , the second term in the R. From Equation 26 , this can be written as. The above equation obeys Weber's law for reward magnitude perception, resulting from errors in both the measurement of magnitude and the measurement of the infinitesimal delay to an immediate reward.
Thus, we predict that even within an individual, the Weber fraction in the perception of reward magnitude subjective value of an immediate reward can change depending on the context, as the past integration interval changes.
The direction of this change will be such that the better the perception of time, the better the perception of reward magnitude.
Further, as mentioned previously after Equation 9 , the above equation also predicts that the larger the experienced reward rate, the larger the error in perception of reward magnitude. These are the strong falsifiable predictions of our account. We now calculate the error in subjective value at a given delay t due to errors in both time and reward magnitude measurement.
From Equation 26 , we get. Grouping the terms that are proportional to SV r, t separately from the other terms, the above equation becomes.
The above equation also abides by Weber's law. Thus, we have shown that the error in subjective value of a given reward delayed by different amounts is proportional to the subjective value at each given delay. Again, the Weber fraction depends on the reward environment of the animal since it depends on r , a est , and T ime. We can also similarly calculate the subjective value error at a given delay for differing reward magnitudes.
This too abides by Weber's law. Thus, we have also shown that the error in subjective value at a given delay for different reward magnitudes is proportional to the subjective value. Previously, we presented a general theory of intertemporal decision-making and time perception TIMERR that explains many well-established observations in these fields Namboodiri et al.
Our theory states that the decisions of animals are a consequence of maximizing reward rates in a limited temporal window including a past integration interval and the delay to a current reward. Interestingly, we showed that the representation of time is also related to the past integration interval in our framework, and that impulsive low tolerance to delays of rewards individuals have an impaired perception of time.
We then demonstrated that the error in perception of time is approximately scalar, with the deviation from exact Weber's law depending on the past integration interval. In this paper, we extend the results of our prior work to consider the role of error in time perception on the perception of reward magnitudes and the subjective values of delayed rewards. We showed that the error in perception of the infinitesimally small delay to an immediate reward affects the perception of reward magnitude in accordance with Weber's law.
This could be the underlying reason behind the observation of Weber's law in the perception of reward magnitude by animals. Subsequently, we showed that in TIMERR, the combination of errors in both time and reward magnitude measurement on the subjective value change of a delayed reward also accords with Weber's law.
Crucially, the Weber fractions are predicted to depend on the reward history of the animal, thus providing a strong, falsifiable prediction of our theory, along with the predicted correlation between errors in time perception and reward magnitude estimation. Superficially, it might be assumed that since the perception of reward magnitude abides by Weber's law, so should the subjective value of a delayed reward.
In fact, such an assertion has previously been made Cui, without the recognition that this requires a specific relation between subjective value, reward magnitude, delay to reward, and the perception of the delay. From our analytical derivation presented above, it is evident that Weber's law in subjective value change is a consequence of the special forms of discounting function subjective value of a delayed reward divided by the subjective value of that reward when presented immediately and subjective time representation that result from our theory.
In fact, if one were to make the standard assumptions of 1 Weber's law in reward magnitude measurement, 2 a hyperbolic discounting function Ainslie, ; Frederick et al. Recent experiments have shown that the representation of reward magnitude or value is not just dependent on the reward under consideration, but also on other available options Huber et al.
A recent neuroeconomic model Louie et al. In light of these findings, one might question our assumption of an absolute code for reward magnitude, i. It is thus important to point out that our theory predicts context dependent choices even under the assumption that the reward magnitude representation is independent of the other available options.
This is because the subjective value of a reward since every reward is effectively a delayed reward is affected by the animal's estimate of its past reward rate [Equation 1 ]. Thus, the presence of distracters affects the subjective value of a reward due to an effect on the past reward rate in experiments involving sequential choices.
Additionally, the current options might affect one's estimate of experienced reward rate Namboodiri et al. Further, as shown in Equations 9, 28 , the larger the value of the past reward rate, the larger the error Weber fraction in representation of a reward. Thus, our theory predicts that the larger the value of the distracter thereby making the past reward rate larger , the higher the errors in deciding between two rewards, in accordance with the experimental observations shown in Louie et al.
The key difference between our account and the divisive normalization account Louie et al. If the brightness needed to yield the just noticeable difference was then the observer's difference threshold would be 10 units i. The Weber fraction equivalent for this difference threshold would be 0.
Using Weber's Law, one could now predict the size of the observer's difference threshold for a light spot of any other intensity value so long as it was not extremely dim or extremely bright. That is, if the Weber fraction for discriminating changes in stimulus brightness is a constant proportion equal to 0. Weber's Law can be applied to variety of sensory modalities brightness, loudness, mass, line length, etc.
The size of the Weber fraction varies across modalities but in most cases tends to be a constant within a specific task modality. This lab will allow the participant to measure their just noticeable difference thresholds for the discrimination of line length using a psychophysical procedure known as the Method of Constant Stimuli.
Choose the longer of the two line segment stimuli presented on the screen for a given trial. You will be asked to enter approximately judgments 60 each at four different levels of standard line size. So when you are in a noisy environment you must shout to be heard while a whisper works in a quiet room. And when you measure increment thresholds on various intensity backgrounds, the thresholds increase in proportion to the background.
Below is a plot of some hypothetical data showing Weber's Law. The slope of the line is the Weber fraction. A TVI plot. Threshold Versus Intensity sometimes called a TVR plot for thresholds for detecting light threshold versus radiance. Weber's Law is not always true, but it is good as a baseline to compare performance and as a rule-of-thumb. On a plot of log I vs log I, the slope of the resulting line is one if Weber's Law holds.
A modified version of Weber's law is as follows:.
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